Sonography techniques use multiple transducer elements for tissue visualization. Signals detected at each element are sampled prior to digital beamforming. The sampling rates required to perform high resolution digital beamforming are significantly higher than the Nyquist rate of the signal and result in considerable amount of data, that needs to be stored and processed. A recently developed technique, compressed beamforming, based on the finite rate of innovation model, compressed sensing (CS) and Xampling ideas, allows to reduce the number of samples needed to reconstruct an image comprised of strong reflectors. A drawback of this method is its inability to treat speckle, which is of significant importance in medical imaging. Here we build on previous work and extend it to a general concept of beamforming in frequency. This allows to exploit the low bandwidth of the ultrasound signal and bypass the oversampling dictated by digital implementation of beamforming in time. Using beamforming in frequency, the same image quality is obtained from far fewer samples. We next present a CS-technique that allows for further rate reduction, using only a portion of the beamformed signal's bandwidth. We demonstrate our methods on in vivo cardiac data and show that reductions up to 1/28 over standard beamforming rates are possible. Finally, we present an implementation on an ultrasound machine using sub-Nyquist sampling and processing. Our results prove that the concept of sub-Nyquist processing is feasible for medical ultrasound, leading to the potential of considerable reduction in future ultrasound machines size, power consumption and cost.

Compressive sensing is the newly emerging method in information technology that could impact array beamforming and the associated engineering applications. However, practical measurements are inevitably polluted by noise from external interference and internal acquisition process. Then, compressive sensing based beamforming was studied in this work for those noisy measurements with a signal-to-noise ratio. In this article, we firstly introduced the fundamentals of compressive sensing theory. After that, we implemented two algorithms (CSB-I and CSB-II). Both algorithms are proposed for those presumably spatially sparse and incoherent signals. The two algorithms were examined using a simple simulation case and a practical aeroacoustic test case. The simulation case clearly shows that the CSB-I algorithm is quite sensitive to the sensing noise. The CSB-II algorithm, on the other hand, is more robust to noisy measurements. The results by CSB-II at $\mathrm{SNR}=-10\,$dB are still reasonable with good resolution and sidelobe rejection. Therefore, compressive sensing beamforming can be considered as a promising array signal beamforming method for those measurements with inevitably noisy interference.